Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 March 2021 doi:10.20944/preprints202103.0279.v1 Article Resolution of the Faint Young Sun Paradox via the Expanding Earth and Radiation Balance Equilibrium Hypothesis Golden Gadzirayi Nyambuya1,2† 0000-0002-3228-6467 1 National University of Science and Technology, Faculty of Applied Sciences — Department of Applied Physics, Fundamental Theoretical and Astrophysics Group, P. O. Box 939, Ascot, Bulawayo, Republic of Zimbabwe; [email protected] 2 The Copperbelt University, School of Mathematics and Natural Sciences — Department of Physics, Theoretical Physics Group, P. O. Box 21692, Jambo Drive — Riverside, Kitwe, Republic of Zambia. 1 Abstract: We present a plausible solution to the now forty seven year old paleoclimatology riddle 2 of the so-called Faint Young Sun Paradox via the combined hypothesis of the conservation of the 3 state of radiation balance between the Earth and Sun and that of an expanding Earth, where, in the 4 face of a changing (increasing) Solar luminosity, the Earth would maintain steady temperatures 5 by re-adjusting the height of its atmosphere. That is to say, depending on whether or not the 6 radius of the solid Earth is changing, this re-adjustment of the height of the Earth’s atmosphere 7 would mean two things — i.e.: (1) either the height increases — in which event the Earth accretes 8 matter from its immediate surroundings (i.e., the obvious pool formed by the Solar wind) thereby 9 increasing the mass of the Earth’s atmosphere, or: (2) the height decreases — in which event 10 the Earth naturally expels matter from its atmosphere, thereby decreasing the effective mass of 11 the Earth. We demonstrate that if — as the current state of the art ITRF observations seem to 12 indicate, namely that — the Earth’s landmass is steadily expanding globally at a paltry rate of −1 13 ∼ +0.45 ± 0.05 mm · yr , and, that the Earth’s atmosphere is to have a present radial vertical 14 height of about one third of the Earth’s radius (∼ 2860 km) from the Earth’s surface, then, one 15 can (might) with relative ease, explain not only the presence of liquid water on the Earth’s surface 16 some ∼ 3.20 ± 0.70 Gyr ago during the Archaean eon when the Sun was about 75% of its current Citation: Nyambuya, G. G. Faint 17 luminosity, but also the present radial expansion rate of the Earth. When all is said and done, Young Sun Paradox. Journal Not 18 the Earth system is herein cast as an auto-self-regulating incubator where the auto-self-regulating Specified 2021, 1, 0. https://doi.org/ 19 mechanism is as a result of the Earth’s atmosphere responding by automatically re-adjusting its 20 height. Received: Accepted: 21 Keywords: Expanding Earth; Exolife; Faint Young Sun Paradox; Radiation Balance Equation. Published: Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional 22 1. Introduction claims in published maps and insti- tutional affiliations. 23 Writing in 1972 in the journal Science, renowned American astronomer, cosmologist, 24 astrophysicist, astrobiologist, author, and one of the greatest inspirational science com- 25 municators in the history of humankind — Carl Edward Sagan (1934-1996), and his Copyright: c 2021 by the author. 26 fellow astronomer — George Mullen; brought (for the first time) to the international lime- Submitted to Journal Not Specified 1 27 light , an apparent paradox [3] concerning the evolution of the Sun and the supposed for possible open access publication 28 presence of liquid water on the Earth’s surface [see e.g. Refs., 3–8]. As initially pointed under the terms and conditions 29 out by Donn et al. [2] and Sagan & Mullen [3] noted that according to the then just of the Creative Commons Attri- 30 established evolutionary stellar models that describe stars like our own Sun — models bution (CC BY) license (https:// 31 that still hold to this day [see e.g. Refs., 9,10]; the Sun’s energy output (which should creativecommons.org/licenses/by/ 4.0/). 1 According to Feulner [1], it was Donn et al. [2] who was were the first — in recorded literature — to point out the apparent discrepancy between the low solar luminosity predicted for the young Sun and the evidence for liquid water on early Earth. Sagan & Mullen [3] where the first to bring this problem to the popular attention of the wider scientific community. Version March 10, 2021 submitted to Journal Not Specified https://www.mdpi.com/journal/notspecified © 2021 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 March 2021 doi:10.20944/preprints202103.0279.v1 Version March 10, 2021 submitted to Journal Not Specified 2 of 20 2 32 have been ∼ 0.75L ) during the Archaean eon [i.e.: tA ∼ (3.80 to 2.50) Gyr ago], should 33 — according to prevalent wisdom of stellar evolution — have been insufficient to sustain 34 liquid water on the Earth’s surface, the meaning of which is that, contrary, to geological 35 evidence [e.g., 4,5], liquid water should not have been present — hence, the ‘paradox’ of 36 how did a Faint Young Sun manage to warm-up the Earth to the extent of sustaining 37 liquid water on its surface despite the ‘insufficient’ energy budget to do so? Accordingly, 38 the Earth should have had only frozen water, thus, making the prospects of the diversity 39 of life witnessed today, very much remote, if not unlikely. This paleoclimatology paradox 40 is today commonly referred to as the Faint Young Sun Paradox (herefater FYS-Paradox) 41 [e.g., 11–23]. 42 For a solution to this apparent riddle, Sagan & Mullen [3] proposed that the Earth’s 43 atmosphere at the time (Archaean eon), must have been rich in carbon dioxide (CO2) 44 and that the consequent greenhouse effect [e.g., 24,25], should have been responsible 45 for warming our nascent planet. However, geological evidence seem to indicate that 46 atmospheric CO2 concentrations during the Archaean and Proterozoic eons were far too 47 low to keep the surface from freezing [see e.g., 5,22,26]. The FYS-Paradox is even more 48 compelling for the planet Mars which we now know to have been covered with oceans 49 for periods of hundreds of millions of years in its nascent life [e.g., 27–29], with only 50 half of the incoming energy flux of Sunlight of the Earth [e.g., 13]. We here make an 51 attempt at this riddle by invoking the hypothesis of an expanding Earth whose albedo 52 is regulated by the radiation balance equilibrium of the Earth system. This supposed 53 radiation balance, we have called it the — Radiation Balance Equilibrium Hypothesis (RBE- 54 Hypothesis). 55 In closing this introductory section, we shall give a synopsis of the reminder of 56 this reading — it is organised as follows: in the subsequent §(2), we discuss some of 57 this proposed solutions to the FYS-Paradox. In §(3), we discuss the expanding Earth 58 hypothesis. In §(4), we present a standard exposition of the derivation of the Solar 59 radiation balance equation. In §(5), we give a critic of this very derivation of the standard 60 Solar radiation balance equation and as a result, we bring in a new concept of the 61 effective radius of the Earth. In §(6), we provide an improved Solar radiation balance 62 equation and we also discuss the concept of a changing albedo for a radially expanding 63 Earth. In §(7), from an effective average global temperature of the Earth standpoint, we 64 discuss the model of the Earth with a constant effective average global temperature and 65 further, we present our proposed solution to the FYS-Paradox. Lastly, in §(8), we give a 66 general discussion and the conclusion drawn thereof. 67 2. Proposed Solutions 68 Proposed solutions to the FYS-Paradox include, but, are not limited to: 69 70 1. Greenhouse Effects [see e.g., 20,21,23,30,31]. In this scenario, we have enhanced 71 greenhouse effect by carbon dioxide or methane, geothermal heath from an ini- 72 tially much warmer terrestrial core, a much smaller Earth albedo, life developing 73 in a cold environment under a 200 m thick ice sheet, a secular variation in the 74 gravitational constant, etc. According to Kasting [22], most of these greenhouse 75 effect models have serious shortcomings: for example, the greenhouse effect from 76 methane appears to be self-limiting, and not enough carbon dioxide is indicated 77 by the geological record to justify a greatly enhanced greenhouse effect in the past. 78 Further, according to Rossing et al. [5], examination of Archaean sediments appears 79 inconsistent with the hypothesis of high greenhouse concentrations. Rossing et al. 80 [5] argues that instead, the obtaining moderate temperature range of the Earth’s 2 Archaean eon, also spelled Archaean eon, is a period in the Earth’s history which began about 4.00 billion years ago with the formation of Earth’s crust and extended to the start of the Proterozoic eon3 2.50 billion years ago. During this time, unicellular organisms are the earliest forms of life that emerged. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 March 2021 doi:10.20944/preprints202103.0279.v1 Version March 10, 2021 submitted to Journal Not Specified 3 of 20 81 system through the eons may be explained by a lower surface albedo brought about 82 by less continental area and the ‘lack of biologically induced cloud condensation nuclei’.